In solid tumors, current limitations in our ability to identify the cell(s) of origin, track the dynamics of tumor subclonal architecture over space and time, and predict drug sensitivities in a mechanistic fashion pose significant challenges to our ability to develop effective treatments and undermine the utility of available therapies, particularly in tumors that become resistant. In this proposal, we argue that approaches based in mathematics and physics are well suited for addressing these critical problems. Historically, the physical sciences have been successful in solving fundamental scientific problems by producing reductionist models that enable the development of conceptual premises, technological breakthroughs that enable new kinds of measurements, and mathematical tools that can accurately represent and predict future activity. In this new Columbia University Physical Sciences-Oncology Center for Topology of Cancer Evolution and Heterogeneity (CUPS-OC), our goal is to develop, validate, and deliver a set of complementary mathematical, technological, and computational approaches that will provide the cancer research community with a framework for unraveling complexity in solid tumors, with the long-term aim of improving diagnosis and treatment. More specifically, the Center's primary research focus will be to study evolution and heterogeneity of solid tumors at the single cell level, by developing lineage tracing techniques in organoid and in vivo systems, single cell sequencing in primary tumors, and emerging genomics-based approaches for identifying targeted therapies. These experimental approaches will be supported by the development of novel, robust, and effective mathematical and computational approaches based in the field of topology, which offers unique and powerful methods for addressing the essential challenge of interpreting the high-dimensional data sets that state-of-the- art genomics technologies generate. In addition, we will promote the integration of physical science approaches into cancer research by both training new researchers and engaging established researchers working at the intersection of the physical sciences and cancer biology. Each of the projects and cores will address technical, mathematical, and biological challenges, contributing to the global enterprise of cancer research and to the development of a broader network of interdisciplinary researchers committed to advancing the field. If successful, we will be able to provide the scientific community with experimentally validated geometric and topological structures of causal inference of clonal evolution, single cell genomic protocols for fast and reliable uncovering of clonal heterogeneity, experimentally validated machine learning approaches for predicting drug sensitivities, and a strong multi-institutional, interdisciplinary program that creates bridges between researchers in pure mathematics, the technology sector, and cancer research.